Angular contact ball bearing



Dec. 30, 1952 FRENKEL 2,623,796

ANGULAR CONTACT BALL BEARING Filed July 21, 1948 jg fer r622 d PatentedDec. 30, 1952 UNITED STATES PATENT OFFICE ANGULAR CONTACT BALL BEARINGMeyer Frenkel, London, England Application July 21, 1948, Serial No.39,862

In Great Britain March 4, 1946 Claims.

This application is a continuation in part of the application Serial No.732,026, filed March 3, 1947, and now abandoned.

This invention relates to angular contact ball bearings. Its object willbe understood from the following consideration.

I haveprovedin my paper Ball and taper roller bearings, published in theJournal of the Royal Aeronautical Society, London, England (No. 423,vol. 50), that in present constructions of angular contact ball bearingsthe forces and couples acting on a ball in the operating bearing produceangular velocities of the ball about three mutually normal axes passinthrough its centre of gravity (that is, spinning and other angularvelocities) the magnitudes of which oscillate among themselves. Thiscauses excessive sliding of the balls on their tracks, and, inparticular, periodic impact of the balls on their tracks and on thecage, involving vibrations and often breakage of the cage, the saidphenomena increasing in severity with rising speed of rotation of thebearing and being largely responsible for the fatigue-effects andgenerally rapid wear, which impose the present known limits on themaximum speeds of operation and lengths of life of angular contact ballbearings, making these incapable of being used at high speeds, withthrust loads.

In many applications, one must, for example, for the above reasons use anumber of radial ball hearings in series to take up a thrust load athigh speed, instead of using one angular contact ball hearing, whichleads to excessive weight and produces troubles, particularly inaeroengines.

The object of this invention is therefore, to provide constructions ofangular contact ball hearings, in which the forces and couples acting onthe balls in the operating bearing, instead of producing the abovetroubles, become useful, preventing the harmful eliects such asoscillations of the angular velocities of the'balls, excessive sliding,impact and the like, from arising at higher speeds, and also causin astate of true rolling (not pure rolling) of the balls on the tracks,thereby enabling angular contact ball bearings to be used for anyrequired high speeds and for large thrusts, for which present angularcontact ball bearings cannot be used, and for great lengthsof life.

in cases, for example, which at present a number of radial ball bearingsare used to take up a thrust load at high speed, the invention thus aimsat providing a construction of angular contact ball bearing replacingthese by one hearing,

2 thus saving weight, which is of particular importance in aero-engineconstruction.

My theory published in my paper in the Joinnal of the Royal AeronauticalSociety gives the unified picture of the occurrences in angular contactball bearings producing the troubles described above, in place of theunrelated fragments of theory hitherto known, and shows the relationsbetween the conditions causing the said troubles, and in this way alsothe relations between the conditions required to prevent the troubles.

These, clothed in mathematical form, lead to a system of equationsrelating among themselves all the dimensions of an angular contact ballbearing, and this leads to the constructional provisions of thisinvention, as described in the following with reierence to theaccompanying drawings.

In these accompanying drawings, Fig. 1 is a half-section through oneembodiment of an angular contact ball bearing in accordance with theinvention, showing also vector-diagrams with reference to which theoperation of the invention is described, and Fig. 2 is a detail of Fig.1 on an enlarged scale. I

In Fig; 1, the ball II is shown interposed between an inner bearing ringI2 having trackgroove l5, and an outer bearing ring I3 havingtrack-groove It, the outer bearing rin l3 being fixed to rotating shaftM, and thus being the driving ring. The profiles of the saidtrackgrooves l5 and it? are respectively symmetrically shaped about axesA2CF2 and AlCFl. The enlarged view of the embodiment in Fig. 2, in whichlike numerals denote like parts, shows the centres of curvature F2 andF1 of the groove-profiles i5 and i6 respectively, which are circular inthis example, F2 and F1 connected to the centre C of the ball llinterposed between the grooves fixing the respective axes of symmetrywhich, according to the invention, enclose an angle less than at C andwhere the axis of symmetry AlCFl of the outer grooves profile isangularly displaced from the axis of symmetry .A2CF2 for the innergroove-profile towards the direction of thecentriiugal force on'theball, as seen from Fig. 1, showing angle a1 less than angle :12, andmore particularly from the triangle of forces taking into account theeffect of centrifugal force Ci. For other shapes of groove-profile, theaxes these are symmetrically shaped about are those respectivelycoinciding with the positions of the radii of curvature of the profiles,which have extreme (maximum orminimum) values, and

pass through the centre of the ball interposed between opposing profilesin the unloaded bearing; or in other words, each profile is such that astraight line passing through the centre of the ball interposed betweenthis profile and the opposing one in the unloaded bearing andintersecting the mid-point of each of those chords of one of saidprofiles, which are perpendicular to said straight line and end in thecontact-area between said one profile and said ball, forms the only axisof symmetry of said one profile.

The acute angle formed between such axes of symmetry A1CF1 and AzCFz,denoted by 9, is related, according to this invention, to the leadingdimensions of the bearin by the condition and 0.2 is the angle ofcontact of the bearing, which is the angle between the rolling-axis ofthe ball, CO, and the bearing axis 00, the rolling axis being the linethrough the ball-centre C and normal to the above-defined axis ofsymmetry of the groove-profile in the non-driving track.

Fig. 2 further shows an inner ring it of the cage or ball-separator.

Other quantities shown in Figs 1 and 2 in connection with theconstructional provisions of the invention to be described in thefollowing, are:

B2A2E2 is the length, in the plane containing the bearing axis, of thecontinuous contact-area under load between the ball i i and the inner,non driving track l5. Half of this length, BzAz or A2E2 subtends anangle 00 at the ball-centre C as indicated. Similarly, B1A1E1 is thelength of the contact area under load between the ball and the outer,driving, track-groove l5, half of which, AiBi or AiEi subtends an angle@x at the ballcentre C, as indicated.

These contact-areas between the ball and its appurtenant tracks areformed under the resultant pressure-forces N1 and N2 on the ball fromthe outer and inner tracks respectively. The force N2 from the innertrack is due to the radially directed load component 10 and to theaxially directed loadcomponent t on the ball in dues tion, as seen fromthe triangle of forces at A2 in Fig. 1, and is here shown to enclose theangl a2 equal to the angle of contact of the bearing with the normal tothe bearing axis. The force N1 from the outer track is due to the axialload component t, to the radial load component w and to the centrifugalforce C: on the ball, as shown in the triangle of forces at A1, andencloses the angle a1=a2--5 with the normal to the bearing axis in thecase shown.

The dimensions of these ccntact-areas under load are determinedaccording to the theory of elasticity, for example from the relevanttheory of elastic deformation by Heinrich Hertz, from the resultantpressure forces acting and from the constructed radii of curvature ofthe grooveprofiles in planes containing the bearing axis. In the exampleshown in Figs. 1 and 2, in which these profiles are circular, these arerespectively the radius of curvature A1F1=p1 for groove it, and AzFz zfor groove 55, although this invention also includes groove-profiles ofother shapes symmetrically shaped about axes, the radius of curvature ofthe contour which has an extreme value and thus defines the said axis ofsymmetry, being hereafter referred to as the central radius of curvatureof the groove-profile.

Conversely, when according to the conditions of this invention, certainlengths of contact areas, in the planes containing the bearing axis andthe centre of the ball in question, are to arise under load, one isenabled from the theory of elasticity to determine accordingly thecentral radii of curvature of the respective groove-profiles which arerequired to bring about such lengths under load.

A further provision of this invention, limiting the constructed centralradii of curvature p1 and p2 of the opposing groove-profiles in relationto the ratio of the pitch-circle radius R to the ball radiusr and to theangle of contact :12 of the bearing, is given with reference to Fig. 2by the following conditions:

The lengths A1B1 and A2132, in the plane containing the bearing axis, ofhalf the contact areas under load between the ball and the driving andnon-driving tracks respectively, which lengths each determine therespective central radius of curvature of its appurtenant groove profilethrough the known functions of the theory of elasticity, are eachrespectively smaller than the corresponding lengths of periphery A1131and AzDz of the ball out off by the, lines A10 and A 0- whichrespectively connect the base-points A1 and A2 of the axes of symmetryA1CF1 and A2CF2 of the groove-profiles in the driving and non-drivingbearing rings with the point 0 on the bearing axis at which the rollingaxis C0 of the ball, through the ball centre C and normal to the axis ofsymmetry A2CF2 of the groove-profile in the non-driving bearing ring,intersects the said bearing axis. Thus the angle subtended at the ballcentre; by the cutoff length of periphery being 2m, i. e. from thegeometry of Fig; 1,

where an le a0 is further determined from the geometry of Fig. 1 by theratio of the pitchecircle radius to the ball radius r and by the angleof ont ct s f th bearing a cording to Therefore, the above conditionsnecessary for the prevention of the troubles in angular contact ballbearings, through limiting the lengths of the contact areas under loadbetween a ball and its tracks, which leng hs each determine the centralradii of curvature of their appurtenant grooveprofile through the knownfunctions of the theory of elasticity, limits the said central radii ofcurvature p1 and p2 of the opposing groove-profiles in relation to theratio of the pitch-circle radius R to ball radius r and to angle ofcontact :12 of the bearing.

Further according to the invention, the constructed central radii ofcurvature p1 and p2 of the opposing groove-profiles are each determinedas function of the leading dimensions R, 1' and angle of contact (12 bythe condition, essential for the prevention of the troubles in angularcontact ball bearings, that the following equations, containing thedependent variables 90, 9x, an, (1x, and 2, and apart from these theleading dimensions R, r and 0.2 of the bearing, determine the angles onand 9x and thereby the required lengths of the contact-areas betweenballs and track-grooves under load, as functions of the leadingdimensions R,1' and m2 of the hearing. The equations, are:

R sin (1 1" tan om where ac is related to angles 90, 9x and on by sind a(l+cosd) (31 13341119 Tr; (1+cos 19) 1+ sin (Cg-17) sin (1 cos 1.)

tan 0 sin 6 sin (vi -19) with 61 and 52 the coefficients (lever arms) ofrolling friction between the ball and the outer and inner tracksrespectively. In this system ax and 9 are further quantities and andwith i the diametral radius of gyration of the ball.

The angles 60 and 9x having thereby been determined as function of theleading dimensions R, r and as: the known functions of the theory ofelasticity, e. g., according to the Hertzian theory, enable the centralradii of curvature p1 and p2 of the respective groove-profiles, requiredto bring about these lengths of contact areas under load, to bedetermined, thereby determining the central radius p1 as well as radiusp2 of curvature of the groove-profiles as functions of the ratio of thepitch-circle radius R, to the ball radius 1' and of the angle of contacte2.

Further, in order to prevent the troubles in angular contact ballbearings, the ratio of pitchcircle radius R to ball radius 1' isdetermined as function of the external load on a bearing and of itsspeed of rotation, as well as of the angle of contact as of the bearingand the acute angle 9 6. enclosed between the axes of symmetry ofopposing groove profiles, according to R 2 t k d k Sin az+2 COS dg]m+tan "2" S111 a where t is the axial component of the load on the ball inquestion, and Cr is the centrifugal force on a ball. According to thisinvention, the angle of contact an and the constructed angle 9 arefurther limited by the condition t (1,.(1 cos a .SlIl a s1ll (a d)$ '2in which [L5 is the coefficient of sliding friction between the ball andits track-surface.

In cases, in which the factor R 2 t k 7 [C, sin 1m The equations showthat in each case a range of external loads and of speeds of rotation iscovered.

The main condition, from which spring all other conditions and theconstructions necessary to prevent the troubles in angular contact ballbearings at high speeds, is that the forces and couples acting on a ballin the operating bearing should give the ball such an angular velocity,that the velocity vector Ve of the ball centre of gravity, and thecouple of the tangential friction forces from the tracks, which servesto overcome the rolling resistance of the ball and with changing shaftspeed to change its rolling angular velocity, and which is coplanar withvelocity vector Vc, should always lie in one and the same diametralplane of the ball, so that the ball should always have the same rollingaxis.

If this is not the case, the said couple due to the friction forces fromthe tracks will develop angular velocities in different diametral planesof a ball, which leads, as proved, to the troubles in angular contactball bearings already described.

The velocity vector Vc of the ball centre of gravity, describing itscircular path about the bearing axis 00, has the same angular velocitywe with reference to the axis CC through the ball centre and parallel tothe bearing axis, as the centre of gravity C has in its circular motionabout the bearing axis 00, and thereforeas seen with reference to Fig.l-the velocity vector V0, and thus also the rolling couple co-planarwith it, has the angular velocity w sin a2 about the axis A20. Hence, inorder that velocity vector V0 and the rolling couple co-planar with itshould always lie in the same diametral plane of the ball, the ball musthave the same angular velocity we sin a2 about axis AzC. If, as shown onFig. l, the resultant angular velocity vector of the ball in the planecontaining its centre C and the bearing axis 00 is w, represented byline CJ enclosing the angle 0:0 with the rolling-axis CD, then, for thefulfillment of the above condition w sin ao=wc Sill a2 and in connectionwith the condition for rolling at point A2, that is that point A2 on theball be instantaneously at rest, making the line A20 connecting A2 withthe point of intersection O 01' the rolling and bearing axes, which isalso at rest, the instantaneous axis of motion of the ball, one obtainsthat the angle a between the resultant angular velocity vector 0.: andthe rolling axis must be (pi 126C. In connection with my theory relatingto the instanteous axis of change of motion of the ball (given in mypaper), the condition 1?. t d k 7' an ---111 a2 m results, requiring theconstructional provision, that the axes of symmetry of opposinggrooveprofiles intersect to form acute angle 2.

Considering the relative motion in the contactarea between the ball andthe non-driving track l5, with reference to Fig. 2, it is seen thatpoints on the ball I I below the instantaneous axis of motion A2D2O,about Which the ball instantaneously rotates, have velocities relativeto the fixed track in the direction opposite to the velocity vector Vcof the ball centre of gravity C, and therefore the sliding frictionforce on the ball from the contact area falling between A2 and D2 is inthe direction of V0. Similarly, all points outside A2D2 on the ball haverelative velocities in the direction of Va, and therefore frictionforces from contact-area falling beyond A2 or D2 are opposite to V0. Thecontact-area between ball and track, however, is symmetrical about pointA2, and therefore, as

long as it does not extend beyond point D2, the

sliding friction forces from the portions of the contact area to bothsides of A2 are opposite and equal, giving no re sultant parallel to Vo,but only a friction couple acting on the ball. However, if thecontact-area between ball and track were to extend beyond D2, thefriction forces from the parts of the contact-area outside A2D2 willexceed those from the parts between A2132, so that a resultant frictionforce on the ball opposite to V0 arises, which at constant shaft speedwould form a couple on the ball changing its angularvelocity about axisAzCz, thereby causing, spin of the ball with all its attendant troubles.The condition to prevent this is that half the length of contact-areaunder load at the non-driving track, and from a similar considerationthat for the driving track A,B, A,D, 11 31 441131 or, in angles arise,and the other conditions claimed in claim 4, which, through the knownfunctions of the theory of elasticity, determined the constructedcentral radii of curvature p1 and p2 of the opposing groove-profiles inthe bearing rings, as functions of the. leading dimensions, of thebearing, namely the ratio and the angle of contact 112 of the hearing.

In connection with the condition that the load on the bearing and thecentrifugal force acting on the balls must produce the forces andcouples on a ball to give it the motion required, the conditions claimedin claims 5 and 6 arise, determining the leading dimensions of thebearing as function of the ranges of external load on the bearing andits speed of rotation.

I claim:

1. A ball bearing comprising two bearing rings, each having a groovedtrack, and a set of balls interposed between said grooved tracks witharea-contact at each track, the mid-point of the area of contact of acall with one track being mor remote from the axis of the assembledbearing than the mid-point of the contactarea of said ball with thesecond track, each of said grooved tracks having a profil in a planecontaining the axis of the assembled bearing and the center of a ball,such that, for each one of said profiles, a straight line passingthrough the center of said ball in the unloaded bearing and intersectingthe midpoint of a chord of said one profile, which is perpendicular tosaid straight line and ends in the contact-area between said one profileand said ball, forms the only of symmetry of said one profile, the saidaxes of symmetry of the opposing profiles in the same plane through theaxis of the assembled bearing enclosing an angle less than and the axisof symmetry of the profile of said one track being angularly displacedwith respect to the axis of symmetry of the profile of the said secondtracl: towards the direction of the centrifugal force on said ball.

2. A ball bearing as claimed in claim 1 in which the said axes ofsymmetry of the opposing groove-profiles form an acute angle 2, thetangent of half of which, in any plans containing the bearing axis,multiplied by the ratio of the pitch-circle radius R to the ball radius7 substantially equals the ratio of the square of the diametral radiusof gyration i of the ball to the sum of the squares of the said radiusof gyration 1 and of the radius r of the ball, multiplied by the sine ofthe angle of contact 0.2 of the bear ing, enclosed in the said plane ofreference between the normal to the above-defined axis of symmetry ofthe groove-profile in the non-driving track the bearing axis, inaccordance with the non-driving and driving tracks respectively whichlengths each determine the respective central radius of curvature of itsappurtenant 9 groove-profile through a known function-are determined asfunction of the ratio and of the angle of contact 0.2 of the bearingfrom the following equations R sin c 1 tan a wher quantity a is definedin relation to angles 00, 0x and as by tan a 0 :1. tan

sin (dz-'17) sin a cos :9

sin (dz-)] sin a;

wher

vi- 2 sin 6, 4

o 2 2 U0 r. tan 9 Z 51 with 61 and 62 the coeflicients of rollingfriction nu between the ball and the said first and second trackrespectively, and in which (1x and :6 are further quantities and i isthe polar radius of gyration of the ball.

4. A ball bearing as claimed in claim, 1 in which the ratio of thepitch-circle radius R to the ball radius r are determined as function ofthe angle of contact 0.2 of the bearing, of the acute angle 9 formedbetween the axes of-symmetry of opposing groove-profile in a planecontaining the bearing axis, and of the range of and 10 external loadson the bearing and of its speeds of rotation, according to the equationin which t is the axial load on a ball in question and C: thecentrifugal force acting on the ball, and

with i the polar radius of gyration of the ball, the angle of contact:12 and the angle 9 of the bearing being further limited by thecondition C, 1 cos a sin a sin (a -1H5 in which as is the coefficient ofsliding friction between the ball and its track-surface.

5. A ball bearing as claimed in claim 1 in which the ratio of thepitch-circle radius R, to the ball radius r being determined as functionof the angle of contact 0.2 of the bearing and of the range of externalloads on the bearing and of its speeds of rotation, according to theequation 7*(6, sin 0J )1+k in which t is the axial load on a ball inquestion, C: the centrifugal force acting on the ball, and

with z the polar radius of gyration of the ball.

MEYER FRENKEL.

REFERENCES CITED The following references are of record in the file ofthis patent:

UNITED STATES PATENTS Number Name Date:

1,769,933 Arutunofi July 8, 1930 1,931,871 Large Oct. 24, 1933 2,102,952Hellyar Dec. 21, 1937 2,316,449 Parker Apr. 13, 1943

